3.977 \(\int \frac{1}{(a+i a \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{3 i}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{i}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{3 i \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{4 \sqrt{2} a \sqrt{c} f} \]

[Out]

(((3*I)/4)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(Sqrt[2]*a*Sqrt[c]*f) - ((3*I)/4)/(a*f*Sqrt[
c - I*c*Tan[e + f*x]]) + (I/2)/(a*f*(1 + I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])

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Rubi [A]  time = 0.175665, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {3522, 3487, 51, 63, 206} \[ -\frac{3 i}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{i}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{3 i \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{4 \sqrt{2} a \sqrt{c} f} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]),x]

[Out]

(((3*I)/4)*ArcTanh[Sqrt[c - I*c*Tan[e + f*x]]/(Sqrt[2]*Sqrt[c])])/(Sqrt[2]*a*Sqrt[c]*f) - ((3*I)/4)/(a*f*Sqrt[
c - I*c*Tan[e + f*x]]) + (I/2)/(a*f*(1 + I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]])

Rule 3522

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Sec[e + f*x]^(2*m)*(c + d*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0] && IntegerQ[m] &&  !(IGtQ[n, 0] && (LtQ[m, 0] || GtQ[m, n]))

Rule 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(a+i a \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}} \, dx &=\frac{\int \cos ^2(e+f x) \sqrt{c-i c \tan (e+f x)} \, dx}{a c}\\ &=\frac{\left (i c^2\right ) \operatorname{Subst}\left (\int \frac{1}{(c-x)^2 (c+x)^{3/2}} \, dx,x,-i c \tan (e+f x)\right )}{a f}\\ &=\frac{i}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{(3 i c) \operatorname{Subst}\left (\int \frac{1}{(c-x) (c+x)^{3/2}} \, dx,x,-i c \tan (e+f x)\right )}{4 a f}\\ &=-\frac{3 i}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{i}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{1}{(c-x) \sqrt{c+x}} \, dx,x,-i c \tan (e+f x)\right )}{8 a f}\\ &=-\frac{3 i}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{i}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}+\frac{(3 i) \operatorname{Subst}\left (\int \frac{1}{2 c-x^2} \, dx,x,\sqrt{c-i c \tan (e+f x)}\right )}{4 a f}\\ &=\frac{3 i \tanh ^{-1}\left (\frac{\sqrt{c-i c \tan (e+f x)}}{\sqrt{2} \sqrt{c}}\right )}{4 \sqrt{2} a \sqrt{c} f}-\frac{3 i}{4 a f \sqrt{c-i c \tan (e+f x)}}+\frac{i}{2 a f (1+i \tan (e+f x)) \sqrt{c-i c \tan (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 1.61837, size = 117, normalized size = 0.94 \[ -\frac{i e^{-2 i (e+f x)} \left (e^{2 i (e+f x)}+2 e^{4 i (e+f x)}-3 e^{2 i (e+f x)} \sqrt{1+e^{2 i (e+f x)}} \tanh ^{-1}\left (\sqrt{1+e^{2 i (e+f x)}}\right )-1\right ) \sqrt{c-i c \tan (e+f x)}}{8 a c f} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]),x]

[Out]

((-I/8)*(-1 + E^((2*I)*(e + f*x)) + 2*E^((4*I)*(e + f*x)) - 3*E^((2*I)*(e + f*x))*Sqrt[1 + E^((2*I)*(e + f*x))
]*ArcTanh[Sqrt[1 + E^((2*I)*(e + f*x))]])*Sqrt[c - I*c*Tan[e + f*x]])/(a*c*E^((2*I)*(e + f*x))*f)

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Maple [A]  time = 0.078, size = 102, normalized size = 0.8 \begin{align*}{\frac{2\,i{c}^{2}}{fa} \left ( -{\frac{1}{4\,{c}^{2}} \left ({\frac{1}{-2\,c-2\,ic\tan \left ( fx+e \right ) }\sqrt{c-ic\tan \left ( fx+e \right ) }}-{\frac{3\,\sqrt{2}}{4}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{c-ic\tan \left ( fx+e \right ) }{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}} \right ) }-{\frac{1}{4\,{c}^{2}}{\frac{1}{\sqrt{c-ic\tan \left ( fx+e \right ) }}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c-I*c*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e)),x)

[Out]

2*I/f/a*c^2*(-1/4/c^2*(1/2*(c-I*c*tan(f*x+e))^(1/2)/(-c-I*c*tan(f*x+e))-3/4*2^(1/2)/c^(1/2)*arctanh(1/2*(c-I*c
*tan(f*x+e))^(1/2)*2^(1/2)/c^(1/2)))-1/4/c^2/(c-I*c*tan(f*x+e))^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c-I*c*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.35799, size = 759, normalized size = 6.12 \begin{align*} \frac{{\left (3 i \, \sqrt{\frac{1}{2}} a c f \sqrt{\frac{1}{a^{2} c f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (6 i \, a f e^{\left (2 i \, f x + 2 i \, e\right )} + 6 i \, a f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{1}{a^{2} c f^{2}}} + 6 i\right )} e^{\left (-i \, f x - i \, e\right )}}{4 \, a f}\right ) - 3 i \, \sqrt{\frac{1}{2}} a c f \sqrt{\frac{1}{a^{2} c f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (\frac{{\left (\sqrt{2} \sqrt{\frac{1}{2}}{\left (-6 i \, a f e^{\left (2 i \, f x + 2 i \, e\right )} - 6 i \, a f\right )} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt{\frac{1}{a^{2} c f^{2}}} + 6 i\right )} e^{\left (-i \, f x - i \, e\right )}}{4 \, a f}\right ) + \sqrt{2} \sqrt{\frac{c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}{\left (-2 i \, e^{\left (4 i \, f x + 4 i \, e\right )} - i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{8 \, a c f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c-I*c*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e)),x, algorithm="fricas")

[Out]

1/8*(3*I*sqrt(1/2)*a*c*f*sqrt(1/(a^2*c*f^2))*e^(2*I*f*x + 2*I*e)*log(1/4*(sqrt(2)*sqrt(1/2)*(6*I*a*f*e^(2*I*f*
x + 2*I*e) + 6*I*a*f)*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(1/(a^2*c*f^2)) + 6*I)*e^(-I*f*x - I*e)/(a*f)) - 3
*I*sqrt(1/2)*a*c*f*sqrt(1/(a^2*c*f^2))*e^(2*I*f*x + 2*I*e)*log(1/4*(sqrt(2)*sqrt(1/2)*(-6*I*a*f*e^(2*I*f*x + 2
*I*e) - 6*I*a*f)*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(1/(a^2*c*f^2)) + 6*I)*e^(-I*f*x - I*e)/(a*f)) + sqrt(2
)*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))*(-2*I*e^(4*I*f*x + 4*I*e) - I*e^(2*I*f*x + 2*I*e) + I))*e^(-2*I*f*x - 2*I*
e)/(a*c*f)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c-I*c*tan(f*x+e))**(1/2)/(a+I*a*tan(f*x+e)),x)

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a \tan \left (f x + e\right ) + a\right )} \sqrt{-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c-I*c*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((I*a*tan(f*x + e) + a)*sqrt(-I*c*tan(f*x + e) + c)), x)